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Question

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

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Solution

Given, AM and DN are the medians of ΔABC and ΔDEF respectively.
To Prove that area(ΔABC)area(ΔDEF)=AM2DN2
Proof
In ΔABC and ΔDEF (Given)
area(ΔABC)area(ΔDEF)=AB2DE2...(i)
[ Because the areas of two similar triangles are proportional to the squares of their corresponding sides.]
and, ABDE=BCEF=CAFD...(ii)
ABDE=12BC12EF=CAFD........(iii)

In ΔABM and ΔDEN, we have
B=E
ABDE=BMEN [from (iii)]
ΔABM and ΔDEN [By SAS similarity criterion]
ABDE=AMDN...(iii)

area(ΔABC)area(ΔDEF) =AB2DE2=AM2DN2.Hence proved.

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